Universal Scaling Invariant of Discrete Relational Networks and its Rigorous Mapping to Chameleon Scalar-Tensor Gravity

We present a rigorous numerical and theoretical verification of a universal scaling law emerging in three-dimensional toroidal networks with non-local interactions of range λλ. Under the critical spatial condition L/λ=4L/λ=4, we compute the edge-averaged Forman-Ricci curvature ⟨κe⟩⟨κe⟩ as a computationally efficient topological proxy. We demonstrate that the normalized curvature density per link, defined as IFR=∣⟨κe⟩λ2∣/⟨k⟩IFR=∣⟨κe⟩λ2∣/⟨k⟩, asymptotically stabilizes towards a finite geometric plateau (I∞≈3.63I∞≈3.63) in the thermodynamic limit. This discrete invariant is mapped via a systematic geometric form factor to the continuum non-minimal coupling ξ=1/6ξ=1/6 of Chameleon Scalar-Tensor Gravity (CSTG). Identifying λλ with the electron Compton wavelength yields a novel geometric derivation of the electroweak scale and the electron Yukawa coupling. Cosmologically, the model introduces a dynamical order parameter η(z)η(z) that naturally resolves the H0H0 and S8S8 tensions (Δχ2≈11.5Δχ2≈11.5 over ΛΛCDM), while predicting a scalar breathing mode in gravitational waves (hb/h+≈0.15hb/h+≈0.15) testable by LISA. The framework provides a concrete bridge from discrete relational networks to continuum gravity and particle physics.